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Difficulty: MediumAbsolute Value Equations and Inequalities

If ww is a real number such that ∣3−2w∣≤9|3 - 2w| \le 9, what is the complete range of possible values for ww?

  1. A
    w≥−3w \ge -3
  2. B
    w≤−3w \le -3 or w≥6w \ge 6
  3. −3≤w≤6-3 \le w \le 6Answer
  4. D
    −9≤w≤9-9 \le w \le 9
  5. E
    −6≤w≤3-6 \le w \le 3

Answer

−3≤w≤6-3 \le w \le 6
The inequality ∣3−2w∣≤9|3 - 2w| \le 9 is equivalent to the compound inequality −9≤3−2w≤9-9 \le 3 - 2w \le 9. Subtracting 33 from all parts yields −12≤−2w≤6-12 \le -2w \le 6. Dividing all parts by −2-2 and reversing the inequality signs results in 6≥w≥−36 \ge w \ge -3, which can be rewritten as −3≤w≤6-3 \le w \le 6.

Step-by-Step Solution

1
Set up the compound inequality representing the absolute value inequality.
−9≤3−2w≤9-9 \le 3 - 2w \le 9
An absolute value inequality of the form ∣f(x)∣≤c|f(x)| \le c is equivalent to −c≤f(x)≤c-c \le f(x) \le c.
2
Subtract 33 from all three parts of the inequality to isolate the term containing ww.
−12≤−2w≤6-12 \le -2w \le 6
To solve for ww, we must isolate the variable term by performing inverse operations on all parts of the inequality.
3
Divide all three parts of the inequality by −2-2 and reverse the inequality signs.
6≥w≥−36 \ge w \ge -3, which is equivalent to −3≤w≤6-3 \le w \le 6
Dividing an inequality by a negative number requires reversing the direction of the inequality signs to preserve the truth of the statement.

Key Concept

Solving absolute value inequalities of the form ∣ax+b∣≤c|ax + b| \le c by converting them into compound inequalities and solving for the variable while reversing the inequality signs when dividing by a negative number.
Estimated Time:1m 30s
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Topics

Absolute Value Equations and Inequalities

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